ACT Form 71H Math Answer Explanation



2014 April(71H) Math Question 1

Word Problems

- Fixed Cost + Variable Cost

1. Danica is trying to encourage her classmates to donate books for the fund-raiser and book-drive for the local library. Danica will donate $35.00, plus $0.07 for each book donated by her classmates. Which of the following equations gives Danica's donation, d dollars, if b books are donated by her classmates?

Choice A

it says that "Danica will donate $35.00, plus $0.07 for each book donated by her classmates.", and that her classmates donated b books:

d = 35 + 0.07b


Choice K

z = -16 × (-4) = 64


Choice D

Rule name Rule Example
Product rules a na m = a n+m 23 ⋅ 24 = 23+4 = 128
a nb n = (a b) n 32 ⋅ 42 = (3⋅4)2 = 144
Quotient rules a n / a m = a nm 25 / 23 = 25-3 = 4
a n / b n = (a / b) n 43 / 23 = (4/2)3 = 8
Power rules (bn)m = bn⋅m (23)2 = 23⋅2 = 64
bnm = b(nm) 232 = 2(32)= 512
m√(bn) = b n/m 2√(26) = 26/2 = 8
b1/n = nb 81/3 = 38 = 2
Negative exponents b-n = 1 / bn 2-3 = 1/23 = 0.125
Zero rules b0 = 1 50 = 1
0n = 0 , for n>0 05 = 0
One rules b1 = b 51 = 5
1n = 1 15 = 1
Minus one rule (-1)n = -1 , n odd v
(-1)n = 1 , n even
(-1)5 = -1
Derivative rule (xn) = nx n-1 (x3) = 3⋅x3-1
Integral rule xndx = xn+1/(n+1)+C x2dx = x2+1/(2+1)+C

4b8⋅5b3 = 4⋅5⋅b8+3=20b11


2014 April(71H) Math Question 4

Angles

- Collinear Points & Triangles

4. In the figure below, ∠QPR measures 40°, ∠PQR measures 105°, and points Q, R, and S are collinear. What is the measure of ∠PRS ?

Choice K

71h

The Exterior Angle Theorem states that, for a triangle:

  • The exterior angle d equals the angles a plus b
  • The exterior angle d is greater than angle a, or angle b.

Therefore, ∠PRS = 40° + 105° = 145°


Choice D

Let the larger number be L, the smaller number be S:

L + S = 90

L = S + 50

⇒ S = 20, L = 70


2014 April(71H) Math Question 6

Absolute Values

- Absolute Value Expressions

6. |7 — 5| — |1 - 8| = ?

Choice G

The absolute value of a number is its distance from zero on a number line.

|7 - 5| - |1 - 8| = 2 - 7 = -5


Choice E

$$3^{2}+2+(-3)^{2}-3=17$$


Choice G

$$(x^{2}+2x)+(3x+5)+(x^{2}+1)+(6x-4)$$ $$=(x^{2}+x^{2})+(2x+3x+6x)+(5+1-4)$$ $$=2x^{2}+11x+2$$


2014 April(71H) Math Question 9

Word Problems

- 2 Variable Addition

8. What is the sum of the 4 binomials listed below? x^2 + 2x, 3x + 5, x^2 + 1, 6x — 4

Choice E

Let the number of Tough Cuts be T, the number of Easy Pushes be E,

T + E = 96

T = 2E

⇒ E = 32, T = 64


2014 April(71H) Math Question 10

Statistics

- Mean, Median

10. Adina plays in a bowling league. Her bowling scores for last week are listed below. What is the median of Adina's bowling scores for last week? 142,186,201,191,116,201,175

Choice H

The median of a set of numbers is the middle number in the set (after the numbers have been arranged from least to greatest) -- or, if there are an even number of data, the median is the average of the middle two numbers.

In descending order, the set of numbers is:

116, 142, 175, 186, 191, 201, 201

Median = 186


2014 April(71H) Math Question 11

Statistics

- Mean, Median

11. The table below shows Shannon’s height, in inches, on her birthday from the day she was born (birth) to age 5. What was the average rate of change in Shannon’s height, in inches per year, from birth to age 5 ?

Choice C

(50 – 20) inches / 5 years = 6 inches


Choice G

The area of a triangle is given by the equation:

Area= (1/2) × base × height

Therefore,

Area= (1/2) × 16 × 12 = 96


2014 April(71H) Math Question 13

Word Problems

- Cost

13. Christopher bought 4 cans of soup for a total of $3.36, which included sales tax of $0.16. At the same per-can cost, what is the cost before the sales tax is added for 6 cans of the same soup?

Choice C

($3.36 – $0.16) ÷ 4 × 6= $4.8


2014 April(71H) Math Question 14

Graphing Properties

- Interpret Graphs or Figures

14. A map of the locations of Marville, Sun Town, and Bright City is shown in the standard (x,y) coordinate plane below, where coordinates for x and y are given in miles. The coordinates of Marville and Bright City are given, and Sun Town is located along a straight line exactly halfway between Marville and Bright City. Highway H from Marville to Bright City is 390 miles long. Highway K from Sun Town to Bright City is 200 miles long. What are the coordinates of Sun Town?

Choice H

71h

Midpoint of a Line Segment
The midpoint is halfway between the two end points:

  • Its x value is halfway between the two x values
  • Its y value is halfway between the two y values

To calculate it:

  • Add both “x” coordinates, divide by 2
  • Add both “y” coordinates, divide by 2

x = (1/2) × (0 + 300) = 150

y = (1/2) × (0 + 160) = 80


2014 April(71H) Math Question 15

Equation of a Line

- Distance & Midpoint

15. The straight-line distance, in miles, from Marville to Bright City must be:

Choice C

71h

Distance between two points P(x1, y1) and Q(x2, y2) is given by:

$$d(P,Q) = \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$$

Distance of a point P(x, y) from the origin is given by:

$$d(0,P) = \sqrt{x^2+y^2}$$

Therefore,

$$d(0,Bright\ City) = \sqrt{300^2+160^2}= 340$$

which is between 300 and 390.


Choice J

71h

390 ÷ 32 × $2.25 = $27.42 ≈ $27


2014 April(71H) Math Question 17

Number Types & Properties

- Order of Values

17. Let 0 <a<b<c

Choice A

For any fraction, the greater its numerator and the lessor its denominator, the greater the value of the fraction.

Among the choices, d / a has the greatest numerator d and the least denominator a.


2014 April(71H) Math Question 18

Word Problems

- 1 Variable Equations

18. Maurice earns $12.50 per hour for the first 40 hours he works each week. For each hour beyond 40 that he works in 1 week, Maurice earns $18.75 per hour. Last week Maurice earned $931.25. How many hours did Maurice work last week?

Choice J

Let the number of extra hours Maurice worked beyond 40 hours be E:

$12.5 × 40 + $18.75 × E = $931.25

⇒ E = 23

⇒ Total hours = 40 + 23 = 63 hours


Choice B

71h

8 × 10 + 12 × 14 + 10 × 12 + 12 × 12 =


2014 April(71H) Math Question 20

Perimeter, Area, Volume

- Other Shapes

20. The perimeter of a parallelogram is 80 inches, and the length of 1 side is 18 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

Choice H

The perimeter of a parallelogram can be found by the formula:

perimeter = 2(w+h)

where:
w is the base length of the parallelogram
h is the side length

80 = 2 (18 + h)

⇒ h = 22

The other sides are: 18, 22, 22.


2014 April(71H) Math Question 21

Trigonometry

- Pythagorean Theorem & sin cos tan

21. The side lengths of right triangle ALMN are given in centimeters in the figure below. What is tan N ?

Choice A

71h

Thus, tan N = 6/11


Choice J

The volume of a right …


Choice A

The perimeter of a rectangle …


2014 April(71H) Math Question 24

Probability

- One Event

24. The figure below shows 3 different views of the same fair cube. Each face of the cube has 1 design drawn on it. The cube will be rolled once. What is the probability that a face showing a star (*) will be on top?

Choice F

The cube has 2 stars …


Choice C

First off, we need to …


Choice F

Plugging the numbers into the …


Choice B

Two triangles are said to …


2014 April(71H) Math Question 28

Angles

- Vertex Angles & Bisectors

28. Distinct lines l and m intersect, forming 4 pairs of adjacent angles. Which of the following statements must be true about these 4 pairs of angles?

Choice K

71h

The measure of a straight …


Choice A

Given any two points on …


2014 April(71H) Math Question 30

Quadratic Equations

- FOIL

30. Which of the following expressions is equivalent to (2x -3)(—x — 7) ?

Choice K

(2x – 3)(-x – 7) …


Choice C

71h

The perimeter of a circle …


2014 April(71H) Math Question 32

Word Problems

- 2 Variable Addition

32. Ursula will order boxes of red pens and boxes of blue pens for her company. The table below gives the number of pens in each box and the price per box. Ursula will order a total of 50 boxes of pens for a total price of $280. Which of the following systems of equations gives a true relationship between the r boxes of red pens and b boxes of blue pens that Ursula will order?

Choice F

71h

Since there are a total …


Choice E

The expression ” the product …


2014 April(71H) Math Question 34

Quadratic Equations

- Factoring

34. The quadratic formula gives the 2 roots for the equation axe + bx + c = O. What are the 2 roots of the equation 3×2 — x = 10 ?

Choice J

The quadratic formula for the …


Choice C

71h

Generally, The area of a …


2014 April(71H) Math Question 36

Graphing Properties

- Create Graphs or Figures

36. The rental fee for the use of a reception hall is $500. To encourage use of the hall’s catering service, the hall’s rental fee is reduced by $100 for each group of 20 meals ordered through the hall’s catering service. All 20 meals in a group must be ordered for each reduction in the rental fee. One of the following graphs shows the hall’s rental fee when 0 to 100 meals are ordered. Which graph is it?

Choice F

It says that “rental fee …


Choice A

Any line parallel to the …


2014 April(71H) Math Question 38

Trigonometry

- Pythagorean Theorem & sin cos tan

38. The figure below shows a 10-foot ladder leaning against a vertical wall. The base of the ladder makes a 65° angle with the level ground. Which of the following expressions gives the distance, in feet, between the base of the ladder and the wall?

Choice G

71h

Let’s refresh the definitions of …


Choice C

Such integers can be:

1, …


Choice G

By definition, each side of …


Choice D

Simplifying the equation, we have: …


2014 April(71H) Math Question 42

Statistics

- Mean, Median

42. Jin's Office Supply sells different styles of notebooks. The sale price and number of sheets per notebook for 3 styles of notebooks are given in the table below. The sale price is the amount a customer pays for that notebook. Which of the following amounts is closest to the average sale price per sheet for a Style B notebook?

Choice F

$1.5 / 125 = $0.012 …


2014 April(71H) Math Question 43

Probability

- Combinations

43. A bin contains 100 Style A notebooks, 100 Style B notebooks, and 100 Style C notebooks. Antoine will select 3 notebooks from the bin. How many different selections of 3 notebook styles are possible? (Note: The order in which the notebooks are selected does not matter.)

Choice C

Let’s divide all the possible …


Choice F

Let the number of Style …


Choice D

200 × 1.6 = 320


Choice J

3x + 5y = 17 …


Choice E

The least common multiple is …


Choice G

The slopes of two lines …


2014 April(71H) Math Question 49

Inequalities

- Number Line

49. Which of the following number line graphs is that of the solution set to the inequality —2x + 12 >= 20 ?

Choice A

Graphing an inequality on a …


2014 April(71H) Math Question 50

Perimeter, Area, Volume

- Other Shapes

50. A right prism with triangular bases and with dimensions given in inches is shown below. What is the prism’s total surface area, in square inches?

Choice G

Total surface area = ½ …


Choice C

1 # 2 = 1 …


Choice G

When a point is translated …


2014 April(71H) Math Question 53

Number Types & Properties

- Complex Number

53. The product of 2 numbers is 25. If 1 of the numbers is the complex number 4 + 3i, what is the other number?

Choice C

What you need to know …


Choice K

71h

What is looks like is …


2014 April(71H) Math Question 55

Angles

- Parallel Lines

55. In the figure below, AB is parallel to DE, the measure of ∠ABC is 49°, and the measure of ∠BCD is 137°. What is the measure of ∠CDE ?

Choice D

71h

When parallel lines get crossed …


2014 April(71H) Math Question 56

Angles

- Circles

56. As shown below, 2 circular pulleys with centers 8 inches apart are connected with a tight belt. The belt wraps 2/3 of the way around the larger pulley, which has a radius of 5 inches, and 1/3 of the way around the smaller pulley, which has a radius of 1 inch. What is the exact length of the belt, in inches?

Choice G

71h

ME = AM – AE = AM – BN = 5 – 1 = 4;

MN = 8

⇒ AB = CD = EN = √(ME² + MN²) = 4√3

arc AMD = 2/3 ⋅ 2π ⋅ 5 = 20/3 ⋅ π

arc BNC = 1/3 ⋅ 2π ⋅ 1 = 2/3 ⋅ π

⇒ Length of the belt = arc AMD + AB + arc BNC + CD = 20/3 ⋅ π + 4√3 + 2/3 ⋅ π + 4√3 = 22/3 ⋅ π + 8√3


2014 April(71H) Math Question 57

Trigonometry

- Graphing Trig Functions

57. Given that sin A = 5/13 and 0° A < 360°, what are all possible values of cos A ?

Choice E

71h

It should be remembered that:

sin2 A + cos2 A = 1

⇒ cos A = ± 12/13

As can be seen from the diagram, in the domain from 0 to 2π (0°-360°), it is possible for cos A to be negative.


2014 April(71H) Math Question 58

Trigonometry

- Advanced sin, cos, tan functions

58. The lengths of the sides of the triangle shown below are given in meters. Which of the following equations gives the degree measure 0 ? (Note: For any triangle, c^2 = a^2 + b^2 — 2ab cos C, where a, b, and c are the lengths of the sides opposite angles with measures A, B, and C, respectively.)

Choice F

We should recognize that the …


2014 April(71H) Math Question 59

Sequences and Patterns

- General Sequences

59. The sum of the first 30 positive integers is 465. Which of the following is the sum of the first 60 positive integers?

Choice D

An arithmetic sequence is a …


2014 April(71H) Math Question 60

Quadratic Equations

- Graphing Quadratic Equations

60. What is the minimum degree possible for the polynomial function whose graph is shown in the standard (x,y) coordinate plane below?

Choice K

The graph crosses the x-axis …